Abstract:The theory of resurgence connects perturbative and non-perturbative physics. In this talk I will demonstrate this connection by focusing on certain one-dimensional quantum mechanical systems with degenerate harmonic minima. I will explain how the resurgent trans-series expansions for the low lying energy eigenvalues, which constitute the semi-classical expansion including non-perturbative terms, follow from the exact quantization condition. In contrast, in the opposite spectral region (with high lying eigenvalues), the relevant expansions are convergent. However, due to the poles in the expansion coefficients, they contain non-perturbative contributions which can be identified with complex instantons. I will demonstrate that in each spectral region there are striking relation between perturbative and non-perturbative expansions even though the nature of these expansions are very different. Notably, the spectra of these quantum mechanical examples encode the vacua of certain N=2 supersymmetric gauge theories in the Nekrasov-Shatashvili limit.